Sets

What is a set? Have you heard the word ‘set’ before? Can you think of sets you might have

at home?

A set is a well-defined collection or group of objects. Golf clubs Teams in the Premiership

Why Sets Are Useful This area of mathematics was created by Georg Cantour in 1874

He was exploring the idea of infinity and wanted a way to analyse his ideas

Set theory forms the basis for a lot of modern mathematics, particularly the language we use and in areas such as;

Software development Artificial Intelligence (Turing Machines)

Writing Something As A Set Use capital letters for the title and lower case letters for the things inside.

Use brackets to show that the set has a beginning and an end

When writing a set we use special brackets called chain brackets { }

Use commas to separate each object

Never repeat an object in a set

Vowels in the English Alphabet {a, e, i, o, u}

Objects In A Set Every object or thing in a set is called an element

If something is an element we use

If something is not an element we use

Defined & Undefined Sets Quickly count the amount of tall people in the class and write the answer in

your copy. Different answers. Why? Based on an opinion.

In the class how many peoples’ name begin with a P All answers the same. Why? Based on a fact.

Equal Sets Describing things as equal is another way of saying that they are the same.

Sets are equal if they contain the same elements. The elements don’t have to be in the same order A = {1, 2, 3, 4} and B = {2, 3, 1, 4} A = B

Defining A Set By A Rule Occasionally the individual elements in a set won’t be written out and

instead they will be described using a sentence or rule. This may be done to save time or space if the set is very large.

C = {x|x is a positive whole number between 1 and 100 inclusive}

The x|x represents the elements

This tells you what elements to fill in

The Null Set The null set is the set that contains no elements

It’s empty Represented by or { }

Is E = {x|x a student older than 16 in this room} a null set? Yes because there are no elements in this set.

Is {0} a null set? No because there is an element within the brackets

Subsets Subsets are simplified versions of a larger set

Each subset must contain elements from the larger or original set We use the symbol to denote a subset or to denote not a subset For example the files on a computer can be written as a subset

Documents OneDrive Documents Willow Park Maths Sets Test

Can you think of other examples of subsets?

Proper & Improper Subsets Proper subsets are simplified versions of a larger set.

These are the type of subsets we’ve been dealing with so far For example {1, 2} {1, 3, 2, 4}

Improper subsets These are subsets that don’t simplify the original set

There are two types; The original set itself The Null set

Cardinal Number When something is written as a set we can investigate lots of things about it

and its elements.

The most basic piece of information is how many elements the set contains. This is the cardinal number of the set

It is represented by # For example if D = {2, 4, 6, 27, 32} Then the #D = 5

Venn Diagrams Another way to represent a set

Usually used to represent multiple sets Each circle represents a separate set A = {1, 2, 3, 4, 5}

A

1 23 45

Intersection A = {1, 2, 3, 4, 5} B = {3, 5, 7, 9}

The intersection contains the elements common to both sets

AB

1 23 5

3 57 9

4

Written like this we break the rule about repeating elements.

AB

1 2

35

7 94

Union Using the same sets as the previous example, we have

Union means bringing things together, so the union of A and B is

AB

1 2

34 5

7 9Note even though 3 and 5 are in both sets we only write them once.

Universal Set This set represents all the elements in a question and so should be the largest set you have. Any

other sets in the questions are subsets. It is represented by a U, written on its own.

This is different to the for the union, which is always written between other letters. In the Venn diagram we represent it using a rectangular box that is drawn around the other sets to

show that it contains everything. U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {1, 2, 3, 4, 5} B = {3, 5, 7, 9}

UA

B

35

Fill in the intersection first.

Then fill in sets A and B

17

4 29

Finally fill in any elements that are left over from the Universal Set

6

8

10

Complement Set This operation excludes a certain set and allows you to analyse whats left

over. It is represented using A’ Using the previous example lets look for A’ (A Complement), or everything not in

A.

UA

B

61 3

72 5

98

4

10

Cover over A and see what’s left

Set Difference Very similar to the complement set, it allows you to exclude a certain set

and analyse what’s left over, but can be more specific than set complement Represented by A \ Using the previous example lets look for A\B (A less B), or everything in A only.

UA

B

61 3

72 5

98

4

10

Looking for A only

Set Problems There are two types of ice cream cones, chocolate and vanilla. You and 24 of

your friends (25 people total) are going to buy ice cream cones. If 15 people buy vanilla, and 20 buy chocolate, how many people bought both flavours?

Draw a Venn diagram and fill in the information you know.

25

CV

2015

There’s a problem. The total that bought ice cream exceeds the amount of people there. This means some people were counted twice.

The people who were counted twice belong to the intersection. They bought both flavours.(20 + 15) – 25 = 10So there are 10 people who were counted twice

Finally clean up the Venn diagram, and check the totals.

25

CV

10 105